The Impact of Sector and Market Variance on

Individual Equity Variance*†

Haoming Wang‡

Professor George Tauchen, Faculty Advisor

* The Duke Community Standard was upheld in the completion of this report

† Honors thesis submitted in partial fulfillment of the requirements for Graduation with Distinction in

Economics in Trinity College of Duke University ‡ Duke University; Durham, North Carolina, 2009

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Acknowledgements

I would like to thank my advisor, Professor George Tauchen, for his help and support during the

completion of this paper. I would also like to thank Professors Bjorn Eraker and Tim Bollerslev

for their help with my research. The comments and suggestions of Brian Jansen, Allison Keane,

Zed Lamba, Michael Schwert, and Caleb Seeley were also integral to the completion of this

thesis.

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Abstract

This paper investigates how changes in measures of sector and market variance affect

equity variance by examining forecasts of equity variance over 1, 5, and 22 day time horizons.

These forecasts were generated using heterogeneous autoregressive regressions that included

measures of sector and market variance. The results demonstrate that sector and market variance

both play an important role in determining equity variance. Further, the inclusion of measures of

sector and market variance improves goodness of fit and decreases forecasting errors. These

results imply that the inclusion of these measures could improve predictive models of equity

variance.

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1. Introduction

The variance of equity returns plays an important role in modern finance. Essential

financial concepts such as derivatives pricing and risk management all depend on forecasts of

equity variance. Recent large intra-day stock price swings have also demonstrated the

importance of using high frequency financial data to calculate variance. This paper will

investigate determinants of equity variance and attempt to create a model that generates more

accurate forecasts of equity variance.

The availability of high frequency financial data has led to the development of predictive

models of variance which utilize intra-day data. One such model is the heterogeneous

autoregressive realized variance (HAR-RV) model developed in Corsi (2003). This model is

used and expanded upon in later sections. The HAR-RV model specifies a stock’s future

variance solely as the function of measures of averages of the stock’s past variances. The

tendency of equity variance to form clusters of high and low variance provides justification for

this specification. This paper proposes that additional information could be realized and the base

HAR-RV model improved by including measures of sector and market variance.

The intuition for this addition is as follows: from popular models such as the Capital

Asset Pricing Model (CAPM), we expect a company’s stock performance to be affected by

factors outside of its own corporate performance. In the case of the CAPM, instead of factors

such as the company’s earnings or its stock’s dividend yield, returns on a market portfolio, the

correlation between the stock and the market portfolio, and the risk free rate determine the

returns on a company’s stock. Even though Fama and French (2004) document the failings of the

CAPM when applied to empirical data, its central tenets still hold: non-company-specific factors

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often play a role in determining the returns on a stock. For example, a hedge fund that is facing

redemptions due to losses in the equity markets might have to sell stock to generate the cash

necessary for meeting redemption requests. This process would depress returns for that stock

regardless of the company’s corporate performance. This leads to a natural extension to equity

variance: if a stock’s returns can be impacted by non-company-specific factors, we expect the

variance of returns to also be impacted by non-company-specific factors. Two such factors are

considered in this paper: the variance of a portfolio constructed from stocks in a company’s

sector, the sector variance, and the variance of a market portfolio, in this case the S&P 500.

The justification for the effect of sector variance is that a company participates in the

same market as their competitors. Thus, it is expected that on average, an increase in the variance

of the sector as a whole, ceteris paribus, leads to an increase in the variance of the stock. These

changes in the variance of stock returns could reflect changing market conditions or any factor

that extends beyond the individual stock and causes uncertainty about the sector. These changes

in market conditions will most likely already be included in longer term measures of the stock’s

variance. However, in the short term it is reasonable to assume that there are sector specific

factors in play because sector wide effects have not yet been fully incorporated into stock returns.

Further, when market-wide variance is high, investors often exit the equity markets and instead

invest in less volatile securities. Thus, we would expect measures of market variance to also

impact stock returns and stock variance.

This paper examines the informational content of measures of sector and market variance

in two specific industries: the pharmaceuticals industry and the banking industry. These two

industries were chosen because they have disparate correlations, or beta factors, with the broader

market and because of the potential to isolate the companies into specific industries. First, in

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sections two and three, the model of asset price movements that is used throughout this paper

and methods to estimate equity variance are introduced. In section three, the base HAR-RV

model is expanded to include measures of sector and market variance. In section four, the data

used throughout the paper is examined and in section five, results of the model with sector and

market variances are described. Section five also introduces re-specification of the model and

discusses the new results that arise.

2. Model of Equity Returns and Variance

2.1. Diffusive and Jump-Diffusive Models

The foundation for our model of equity returns is the following stochastic differential

equation which expresses the continuous logarithmic price path of a stock as

𝑑𝑝 𝑡 = 𝜇 𝑡 𝑑𝑡 + 𝜎 𝑡 𝑑𝑊 𝑡 , 0 ≤ 𝑡 ≤ 𝑇. (1)

In this model, μ(t) represents the time-dependent drift component of the asset price, σ(t)

represents the time dependent volatility of the price, and dW(t) is a standard Brownian motion.

However, Merton (1976) argued that intuitively, stock price dynamics cannot be represented by a

stochastic process with a purely continuous path because we can often observe large

discontinuous stock price movements which appear to be “jumps”. Due to this flaw in the pure

continuous model, Merton added a jump component to the standard diffusive model. Empirical

data support Merton’s assertion: the presence of large, discontinuous price movements in

financial markets caused by new information is an oft observed phenomenon and is discussed in

works such as Andersen, Bollerslev, and Diebold (2006). Under Merton’s jump-diffusion model,

the path of the logarithmic price is expressed as

𝑑𝑝 𝑡 = 𝜇 𝑡 𝑑𝑡 + 𝜎 𝑡 𝑑𝑊 𝑡 + κ 𝑡 𝑑𝑞 𝑡 , 0 ≤ 𝑡 ≤ 𝑇. (2)

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The first two terms of the right side of Equation 2 represent the continuous part of the log price

process and are exactly the same as in the pure diffusive model. The third term accounts for

jumps: q(t) is a counting process (usually assumed to be the Poisson arrival process) and k(t) is

the size of the corresponding discrete jump in the otherwise continuous price process.

2.2. Measures of Variance

The availability of high frequency intra-day financial data has allowed the development

of measures of variance that better reflect the amount of variation in a stock’s price. The

importance of intra-day data is best seen with an example: on October 23rd

, 2008, the S&P 500

opened the trading session at 902.99 and closed at 908.11 for a gain of 0.56%. However, during

the day, the S&P reached a low of 865.24 and a high of 919.45, a trading range of 6%. Using

price data at the daily frequency, or even low frequency intraday data, would lead to us to miss

important information about how volatile the S&P 500 was on October 23rd

.

An important measure of the variance in our model of equity returns is quadratic

variation; however, quadratic variation is a continuous time calculation, and thus must be

estimated instead of directly observed because changes in stock prices happen in discrete time

intervals. One consistent estimator of quadratic variation is realized variance, which is a daily

calculation. We let t be the day and M be the sampling frequency. Throughout the paper samples

are taken at a 5 minute interval, which corresponds with M, the number of daily observations,

equal to 78. Then intraday geometric returns are defined as

𝑟𝑡 ,𝑗 = 𝑝 𝑡 − 1 +𝑗

𝑀 − 𝑝 𝑡 − 1 +

𝑗 − 1

𝑀 , 𝑗 = 1,2,3…𝑀 (3)

where p is the log price. The realized variance is then defined as

8

𝑅𝑉𝑡 = 𝑟𝑡 ,𝑗2

𝑀

𝑗=1

4 ,

which is the sum of squared intra-day log returns. Andersen and Bollerslev (1998) emphasize

that according to the theory of quadratic variation, the above converges to the integrated variance

plus the jump component as the time between observations approaches zero. That is,

lim𝑀→∞

𝑅𝑉𝑡 = 𝜎2 𝑠 𝑑𝑠 + 𝑡

𝑡−1

𝜅2(𝑠)

𝑡−1<𝑠≤𝑡

. (5)

The first term corresponds to the integrated variance of the continuous process and the second

term corresponds to the squared discrete jumps. This limit is also the definition of quadratic

variation, which shows that realized variance is a consistent estimator of quadratic variation.

While the realized variance measures both the variation of the continuous process and the

jump process, the realized bipower variation only measures the variation of the continuous

process. The realized bipower variation is defined as

𝐵𝑉𝑡 = 𝜇1−2

𝑀

𝑀 − 1 𝑟𝑡,𝑗−1 𝑟𝑡,𝑗

𝑀

𝑗=2

=𝜋

2

𝑀

𝑀 − 1 𝑟𝑡 ,𝑗−1 𝑟𝑡 ,𝑗

𝑀

𝑗=2

(6)

with

𝜇𝑎 = E 𝑍 𝑎 , 𝑍~ 𝑁 0,1 ,𝑎 > 0. (7)

By multiplying adjacent returns, the effect of returns that are jumps is mitigated. Bipower

variation is thus a consistent estimator of the integrated variance of the continuous price process,

which means that

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lim𝑀→∞

𝐵𝑉𝑡 = 𝜎2 𝑠 𝑑𝑠𝑡

𝑡−1

. (8)

3. Methods

3.1. Background for HAR-RV Type models

The tendency of equity variance to form into clusters of low and high levels is a common

characteristic of the financial markets and can be seen clearly in Figure 1. Because of this

clustering, even though it is nigh-impossible to predict equity returns, the variance of equity

returns can be forecasted. The generalized autoregressive heteroskedasticity model (GARCH) of

Bollerslev (1986) and the HAR-RV model are two different ways to model variance clustering.

Both models attempt to predict future equity variance based on historical measures of variance.

However, only the HAR-RV model utilizes the additional information found in intraday price

data.

GARCH forecasts variance as a function of past squared residuals and past predictions of

variance. The most popular GARCH specification is the GARCH(1,1) model, which forecasts

today’s variance as a weighted average of yesterday’s variance forecast and yesterday’s squared

residual term. On the other hand, the HAR-RV model predicts that future variance is a function

of past averaged realized variances at different frequencies. The interpretation of this is that each

historical measure of averaged realized variance will play a role in determining expectations of

future variance. Changes in expectations of future variance will in turn cause traders to act,

influencing future realized variance.

Empirical evidence from Andersen, Bollerslev, Diebold, and Labys (2003) has shown

that linear models with historical intra-day variance measures as regressors, such as the HAR-RV

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model, are better predictors of variance than GARCH type models. This is because the GARCH

model only makes use of daily return data, and isn’t able to take advantage of the additional

information found in the intra-day data. Due to the superior performance of HAR-RV type

models, as well as the availability of high frequency intraday data, this paper will use the HAR-

RV model as a foundation for further work.

3.2. HAR-RV Model Specification

HAR-RV type regressions make use of average realized variance over daily, weekly, and

monthly periods. The average realized variance over some h discrete periods is

𝑅𝑉𝑡,𝑡+ℎ =𝑅𝑉𝑡+1 + 𝑅𝑉𝑡+2 + ⋯ + 𝑅𝑉𝑡+ℎ

ℎ (9)

with h =1 corresponding to daily periods, h=5 corresponding to weekly periods, and h=22

corresponding to monthly periods. Thus the standard HAR-RV model is expressed as:

𝑅𝑉𝑡,𝑡+ℎ = 𝛽0 + 𝛽𝐷𝑅𝑉𝑡−1,𝑡 + 𝛽𝑊𝑅𝑉𝑡−5,𝑡 + 𝛽𝑀𝑅𝑉𝑡−22,𝑡 + 𝜀𝑡+1. (10)

The model is used to predict values for day-ahead, 5-days ahead, and 22-days ahead realized

variances. These time horizons correspond to day-ahead, week-ahead, and month-ahead

predictions of average realized variance. Throughout this paper, Newey-West covariance matrix

estimators with a lag of 60 days are used to calculate standard errors. The Newey-West method

for estimating standard errors is a common statistical procedure designed to account for auto-

correlation in the residuals of time series data. The Newey-West method assumes that the

correlation between residuals approaches zero as the time between observations approaches

infinity. The covariance of lagged residuals is also weighted by an inputted maximum lag time

horizon, in this case 60 days. Thus, greater weight is placed on the covariance of observations

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that are closer together. Newey-West estimates of standards errors are necessary given the time-

varying nature of equity variance.

In order to determine the effect of sector and market variance on individual equity

variance, two extra sets of regressors are added to the base HAR-RV model in Equation 10. First,

the sector realized variance measure is constructed as an average of the realized variance of all

companies in the S&P 100 within that sector. For example, the pharmaceuticals sector realized

variance was constructed as the average of the realized variance of Abbott Labs (ABT), Bristol

Myers Squibb (BMY), Johnson & Johnson (JNJ), Merck (MRK), and Pfizer (PFE). The market

realized variance was calculated using S&P 500 futures data, also taken at five minute intervals.

Thus, the first model introduced is of the following form:

𝑅𝑉𝑡,𝑡+ℎ = 𝛽0 + 𝛽𝐶𝐷𝑅𝑉𝑡−1,𝑡 + 𝛽𝐶𝑊𝑅𝑉𝑡−5,𝑡 + 𝛽𝐶𝑀𝑅𝑉𝑡−22,𝑡 + 𝛽𝑆𝐷𝑅𝑉𝑠𝑒𝑐𝑡𝑜𝑟 ,𝑡−1,𝑡

+ 𝛽𝑆𝑊𝑅𝑉𝑠𝑒𝑐𝑡𝑜𝑟 ,𝑡−5,𝑡 + 𝛽𝑆𝑀𝑅𝑉𝑠𝑒𝑐𝑡𝑜𝑟 ,𝑡−22,𝑡 + 𝛽𝑆𝑃𝐷𝑅𝑉𝑚𝑎𝑟𝑘𝑒𝑡 ,𝑡−1,𝑡

+ 𝛽𝑆𝑃𝑊𝑅𝑉𝑚𝑎𝑟𝑘𝑒𝑡 ,𝑡−5,𝑡 + 𝛽𝑆𝑃𝑀𝑅𝑉𝑚𝑎𝑟𝑘𝑒𝑡 ,𝑡−22,𝑡 + 𝜀𝑡+1 (11)

with C denoting the individual company, S denoting sector, and SP denoting market. While this

model is the starting point for this paper, reduced forms of this model will also be examined in

later sections. This model is referred throughout the paper as the Full model.

3.3. Jump Test Statistic

A part of recent literature on financial markets has been focused on developing tools to

detect intraday jumps in asset prices. In this section of the paper we examine statistics that test

for jumps that are taken from Barndorff-Nielsen and Shephard (2004). We can see from the

definition of realized variance and bipower variation that

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lim𝑚→∞

(𝑅𝑉𝑡 − 𝐵𝑉𝑡) = 𝜅2(𝑠)

𝑡−1<𝑠≤𝑡

, (12)

which is the jump component of asset price variance. Thus, a measure of the percentage of total

variance caused by the jump process is the relative jump,

𝑅𝐽𝑡 =𝑅𝑉𝑡−𝐵𝑉𝑡

𝑅𝑉𝑡. (13)

In order to standardize the relative jump into units of standard deviation, any test statistic also

needs to incorporate an estimate of integrated quarticity. One such estimator that is commonly

used is the Tri-Power Quarticity, which is defined as:

𝑇𝑃𝑡 = 𝑀𝜇43

−3(𝑀

𝑀 − 2) 𝑟𝑡 ,𝑗−2

4/3 𝑟𝑡,𝑗−1

4/3 𝑟𝑡,𝑗

4/3. 14

𝑀

𝑗=3

Huang and Tauchen (2005) discovered that in simulations, the best performing test statistic for

jump detection is the Z-Tripower-Max test statistic, which is defined as:

𝑍𝑇𝑃 ,𝑟𝑚 ,𝑡 =𝑅𝐽𝑡

(𝑣𝑏𝑏 −𝑣𝑞𝑞 )1

𝑀max (1,

𝑇𝑃 𝑡𝐵𝑉 2)

, 𝑣𝑏𝑏 − 𝑣𝑞𝑞 = (𝜋

2)2 + 𝜋 − 5, (15)

where ZTP,rm,t has a standard normal distribution. Values of the test statistic that correspond to p-

values at the 0.001 level are flagged as days with jumps.

4. Data

High frequency data for 11 individual stocks in two sectors as well as S&P 500 futures

were examined. The data were purchased from an online data vendor, price-data.com, and

include the pharmaceuticals: Abbott Labs (ABT), Bristol Myers Squibb (BMY), Johnson &

Johnson (JNJ), Merck (MRK), and Pfizer (PFE) as well as the banks: Bank of America (BAC),

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Bank of New York (BK), Citigroup (C), JP Morgan (JPM), US Bancorp (USB), and Wells-Fargo

(WFC). Data for the pharmaceuticals sector were obtained for 10 years, from April 1997 until

the end of October 2007. This corresponds to a total of 2,682 trading days of price data. In-

sample results for the model were calculated from the beginning of the dataset until October

2006, in order to have a year’s worth of data to test the model out-of-sample. Data for the

banking sector are from August 1997 until October 2007. This corresponds to a total of 2,595

trading days of price data. Similar to the pharmaceutical stocks, the last year’s worth of data was

used for out-of-sample testing. In-sample results were again generated using data from August

1997 to October 2006. For all datasets, non-full trading days were removed so that each day

would have the full complement of 78 5-minute observations which were then used to calculate

multiple measures of variance.

5. Results

The preceding sections have been concerned with creating a predictive model of equity

variance incorporating sector and market variances. In the following sections, results for the Full

model are examined. In the course of this examination, it becomes apparent that the Full model

needs to be pared down. After a discussion of the results for the Full model, reduced forms of the

Full model are introduced and discussed.

5.1. In-sample Results (Full Model)

By examining the R-squared measures of both the standard HAR-RV and the Full model,

one can see whether measures of sector and market variance include additional information

about individual equity variance. Since the R-squared is the proportion of variation in the

dependent variable explained by variation in the regressors, models with higher R-squared are

considered to be superior. Further, by examining the coefficients of the regression models, we

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can better understand what determines the variance of individual equities, and how that effect

might change across companies and industries. Coefficient estimates for the pharmaceuticals and

R-squared measures for both industries can be seen in tables 2 through 7.

Both the HAR-RV and the Full model were tested over three time spans: day-ahead

predictions, 5 days-ahead predictions, and 22 days-ahead predictions. As in previous papers such

as Fradkin (2007), we see the highest R-squared for 5 days-ahead predictions, followed by 22

days, and then finally 1 day ahead regressions. This makes sense: the model should be better at

explaining the measures of realized variance that have been smoothed out over 5 or 22 days

because realized variance can undergo large day-to-day changes.

The in-sample results for both the pharmaceuticals and the banks suggest that the

inclusion of measures of realized variance for both the sector and the market provides additional

information for predictions of equity realized variance. Across the board, goodness of fit for the

models is increased. There is an average increase in R-squared of 5.5% for day-ahead predictions,

5.95% for 5 days-ahead predictions, and 5.85% for 22 days-ahead predictions. Further, there is a

wide range in improvements in R-squared, with increases ranging from 17% to 1.6%. The Full

model was also separated into two versions: one version that just included the sector regressors

and one version that just included the market regressors. On average, the R-squared increase

from a model incorporating only the stock and the market to the Full model is greater than the

increase from a model incorporating only the stock and the sector to the Full model. This might

suggest that measures of sector realized variance include more unique information than the

market realized variance. We can also conclude that both the sector and the market realized

variance measures contain unique information because models incorporating only one or the

other have lower explanatory power than models that contain both. One important note is that we

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see these improvements in explanatory power for stocks in both the banking and pharmaceuticals

sectors. For two such structurally different industries to exhibit the same benefits from the

inclusion of measures of sector and market variance suggests that these are wide ranging results

that can be applied to most equities.

Further, the coefficients provide some interesting insights: for almost all stocks, the one

day lagged sector realized variance is statistically significant, positive, and also relatively large.

From this we can conclude that there are clear sector effects that determine equity variance.

Again this makes intuitive sense, we would expect there to a positive relationship between sector

variance and equity variance because we expect some spill-over from factors that would

influence the entire sector. The importance of the lagged daily sector realized variance can be

seen in both the banks and the pharmaceuticals. For such a consistent result to be seen across

these two very different industries, it would suggest that the previous day’s sector variance plays

an important role in determining the variance of most equities. This result provides some

evidence for sector variance influencing stock variance.

However, in general, careful interpretation of these coefficient estimates is required.

Some coefficient estimates are easily interpretable, with the daily lags decreasing in importance

as the predictive time horizon increases and the weekly and monthly lags increasing in

importance. Unfortunately, many of the coefficients are not statistically significant, which is

especially troubling for the lagged values of the realized variance of the stock itself. The

coefficient estimates for the Full model also consistently exhibit a negative coefficient for the

lagged monthly market realized variance. This means that increases in the level of the average

market variance over the last month actually imply lower levels of future equity variance. One

possible reason why this occurs is because the regression coefficient is defined as the predicted

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impact on the stock’s realized variance when the market realized variance rises, while keeping

stock and sector-specific realized variance constant. Thus, this negative coefficient could

demonstrate the tendency of market participants to rotate away from volatile stocks and into

more stable stocks.

Interestingly, this effect is different across the banking and pharmaceutical sectors.

While this negative relationship is statistically significant across the pharmaceuticals, it is no

longer statistically significant with the bank stocks. Further, the absolute values of the negative

coefficients are lower, and in some cases are even positive. This could be because the beta

factors, a measure of the correlation of stock returns to market returns, of both industries are

significantly different. The beta factor of the banking sector is 1.15 while that of pharmaceuticals

is 0.70, meaning that returns from bank stocks are much more correlated with the market than

returns from pharmaceutical stocks. This could lend more evidence to the view that this negative

coefficient is tied to investors rotating their holdings into certain industries. The negative

correlation between equity returns and equity variance is a well-known characteristic of stock

markets and is the reason for implied volatility skew on equity options. Thus, if the variance of

the market portfolio increases, this could mean that on average, the returns on the market

portfolio are decreasing. Because banking stocks are pro-cyclical, returns on banking stocks are

likely to be depressed as well. This leads to selling, which could mitigate the impact of the

rotation interpretation. However, another possibility is that our model needs to be re-specified, a

subject that will be discussed in the next section.

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5.2. Model Shortcomings

One specific aspect of the Full model that makes interpretation of coefficient estimates

difficult is the high correlation between some regressors. There is high correlation between

market and sector variables: the correlation between the S&P monthly realized variance and the

sector monthly realized variance is 0.9298 for the banks and 0.9127 for the pharmaceuticals.

Further, there is high correlation between the sector and market weekly realized variance for the

banks and the pharmaceuticals. There is also high correlation between daily sector realized

variance and daily market realized variance for the pharmaceuticals. Finally, the data also exhibit

high correlation between lagged monthly realized variance for the equity and for the market in

both sectors.

This high correlation between the regressors is known as collinearity. While the presence

of collinearity does not affect the explanatory ability of a regression, it makes it so that

coefficient estimates become unpredictable. This occurs since it is impossible to determine the

individual effects of the regressors on the dependent variable because the regressors are so highly

correlated. Since part of the goal of this paper is to investigate the dynamics of what determines

individual equity variance, collinearity and the associated unreliable coefficients pose a problem.

Because there are highly correlated regressors, re-specifying the original model with new

regressors might solve this collinearity problem.

However, collinearity is not the only reason why re-specification of the Full model might

be necessary. Harvey (1980) and Granger and Newbold (1974) show that in general, regressions

involving economic variables in levels can be misleading and could cause problems with

interpretations. One possible solution suggested by both papers is to use the first differences

18

between variables instead of the levels. Because we have coefficients whose values seem

difficult to interpret, taking differences between related variables could yield more interpretable

results.

5.3. Model Re-specification

The presence of collinearity and the difficult to interpret results of the Full model

motivate the need for re-specification of the Full model. The first step is to determine which

variables are significant in the original model. Multiple F-tests were performed on the Full

model to determine which sets of regressors were jointly insignificant and could thus be dropped

from the model. One pattern emerged: for almost every stock and time horizon combination the

coefficients of the lagged weekly and monthly sector realized variance, as well as the lagged

weekly market realized variance, were statistically insignificant. The p-values of the coefficients

for these regressors can be seen in Table 11. Further, because of the high correlation between the

lagged monthly market and equity realized variances, a new regressor was constructed as the

difference between the lagged daily market realized variance and the monthly market realized

variance. This new regressor has a clear interpretation: it is the daily deviation from the average

realized variance over the last month. A new Parsimonious model was created by taking the Full

model and dropping the lagged weekly and monthly sector realized variance, as well as the

lagged weekly market realized variance. Finally, the lagged daily and monthly market realized

variances were replaced with the measure of their difference. The model is defined as:

𝑅𝑉𝑡,𝑡+ℎ = 𝛽0 + 𝛽𝐶𝐷𝑅𝑉𝑡−1,𝑡 + 𝛽𝐶𝑊𝑅𝑉𝑡−5,𝑡 + 𝛽𝐶𝑀𝑅𝑉𝑡−22,𝑡 + 𝛽𝑆𝐷𝑅𝑉𝑠𝑒𝑐𝑡𝑜𝑟 ,𝑡−1,𝑡

+ 𝛽𝑆𝑃𝐷𝑖𝑓𝑓 (𝑅𝑉𝑚𝑎𝑟𝑘𝑒𝑡 ,𝑡−1,𝑡 − 𝑅𝑉𝑚𝑎𝑟𝑘𝑒𝑡 ,𝑡−22,𝑡) + 𝜀𝑡+1. (16)

19

The intuition behind this model is that the determinants of equity realized variance are the

historical measures of the stock’s realized variance as well as yesterday’s sector variance and the

daily deviation from the monthly average market realized variance.

While this Parsimonious model is mostly robust through both the day-ahead predictions

and the week-ahead predictions, it struggles to match the performance of the Full model for

month-ahead predictions. For the updated month-ahead predictive model, the Parsimonious

model is taken; however, the daily deviation from the monthly mean of market realized variance

is replaced with the lagged monthly level of market realized variance. The lagged monthly level

of sector realized variance is also added back to the model. This replacement causes the problem

of collinearity to reappear, but the difference in explanatory power is too pronounced to ignore.

The third and final model that we use will be referred as the Reduced Monthly model and is:

𝑅𝑉𝑡,𝑡+ℎ = 𝛽0 + 𝛽𝐶𝐷𝑅𝑉𝑡−1,𝑡 + 𝛽𝐶𝑊𝑅𝑉𝑡−5,𝑡 + 𝛽𝐶𝑀𝑅𝑉𝑡−22,𝑡 + 𝛽𝑆𝐷𝑅𝑉𝑠𝑒𝑐𝑡𝑜𝑟 ,𝑡−1,𝑡

+ 𝛽𝑆𝑀𝑅𝑉𝑠𝑒𝑐𝑡𝑜𝑟 ,𝑡−22,𝑡 + 𝛽𝑆𝑃𝑀𝑅𝑉𝑚𝑎𝑟𝑘𝑒𝑡 ,𝑡−22,𝑡 + 𝜀𝑡+1. 17

This model suggests that average monthly equity variance is driven by its own lagged realized

variances, the level of lagged daily and monthly sector realized variance, and the level of the

lagged monthly market realized variance. This need for re-specification makes some sense: we

would expect that the level of these longer time horizon realized variance measures to have

greater impact on the longer time horizon predictions than on the shorter time horizon

predictions.

For the day-ahead predictions, there is an average decrease in R-squared from the Full

model by using the Parsimonious model of only 0.56%. For week-ahead predictions, the loss in

explanatory power is higher, at 1.04%. Using the Parsimonious model for the month-ahead

20

predictions, there is a decrease in R-squared of 2.3%. However, using the Reduced Monthly

model, the percent decrease in R-squared is only 0.56%. The full R-squared results can be seen

in Tables 4-6.

Interpretation of the coefficient estimates of the Parsimonious model is also much clearer

than with the Full model. All coefficients are positive and we have mitigated the problems of

collinearity by introducing the market difference as a regressor. As before, there is a clear pattern

that the coefficient estimate for the lagged daily sector realized variance is both relatively large

as well as statistically significant across almost every time horizon. This also holds in the

Reduced Monthly model. It appears that sector variance is an important factor in individual

equity variance. The coefficient estimate for the daily deviation in market realized variance from

the last month’s average is also always positive. The interpretation of this result is that an

increase in market variance relative to the last month’s average has a positive impact on equity

variance.

The Parsimonious model is remarkable because the Full model has been distilled into a

streamlined version that highlights the critical factors that drive equity variance. This result

provides further evidence for the intuitive guess that sector and market variance impact stock

variance. Moreover, the Parsimonious model highlights that these effects mainly occur in the

short-run. They act in the one day and one week time horizons through yesterday’s sector

variance and yesterday’s deviation in market variance from the average over the last month. That

they act in the short run is to be expected: it seems intuitive that the information contained in

averages of weekly and monthly sector and market variances has already been incorporated into

stock returns. Complete results for the Parsimonious and Reduced Monthly models can be seen

in Tables 11-13.

21

By examining both the Full model and these two new models, we can conclude that there

are sector and market factors which play a role in equity variance. Moreover, the Full model,

which includes all measures of sector and market variance, has been streamlined into more

intuitive versions. This was accomplished by using F-tests to discover coefficients that were

consistently insignificant. It is also striking that across all models and all time horizons, the

lagged daily sector realized variance plays a significant role in determining the variance of a

stock. This demonstrates that while new information about the sector or the market is often

already realized in longer term measures of equity variance, in the short run, there is unique

information contained in the variance of a sector or the variance of the market. Finally, the

inclusion of measures of sector and market realized variances clearly provide more accurate

results when forecasting equity variance.

5.4. Out-of-sample Results

The evolution of realized variance over the full sample period is especially striking, as

seen in Figure 1; there are clearly distinct low variance and high variance regimes. In order to

examine how the models investigated in this paper perform in these two regimes, all models

were tested out-of-sample using data from November 2006 to November 2007. This date range is

particularly interesting because there are two distinct periods of low and high variance. The first

half of the sample is a period of rather low equity variance; however, starting in the summer of

2007, we see marked increases in equity variance due to the subprime mortgage crisis. This will

allow testing of each model in two very distinct environments, similar to the circumstances that

we see in the full sample. Out-of-sample testing was performed separately using each model for

all of November 2006 to November 2007, the first half of the sample, and the second half of the

sample.

22

One clear pattern is that for the earlier sampling period of relatively low equity variance,

models that incorporate sector and market realized variances consistently outperform the base

HAR-RV model. Further, there is a clear difference between how the models perform for the

first half of the data and for the second half: model performance is consistently better during the

lower variance regime. The benefits of including sector and market variance measures in the

model also are much greater in the earlier part of 2007. This effect is especially pronounced for

the banking sector. For the pharmaceuticals sector, on average the mean-squared error (MSE) of

the Full model is 19% lower than that of the base HAR-RV model for the early period while the

Full MSE is only 10% lower for the later period. For the banking sector, on average the MSE of

the Full model is again 20% lower than of that of the base model; however, for the later period,

the MSE of the Full model is actually higher by about 1%. The poor performance of the Full

model is troubling. Since on average the banking sector was more volatile during the credit crisis

than the pharmaceuticals sector and the Full model for the banking sector performed worse than

the base model, this would suggest that the performance of the Full model suffers in periods of

high variance.

This difference in performance is apparent when looking at Figures 2-4, which plots the

residuals and out-of-sample predictions for JP Morgan. There’s a marked increase in the level of

the residuals once the subprime mortgage crisis begins and we enter the high variance regime.

Further, we see one specific spike in variance that the models incorporating market and sector

variance dramatically overestimate. While in-sample performance of the Parsimonious model

was relatively similar to that of the Full model, the out-of-sample performance of the

Parsimonious model is significantly worse, offering much less improvement over the base HAR-

RV model. Even though there is the problem of collinearity with the Full model, it seems that its

23

explanatory ability is much stronger. However, the Reduced Monthly model does perform better

on average.

One interesting dynamic is that for the first half of the year, during the low variance

regime, the models all consistently over-predict equity realized variance. This is especially

obvious for the week and month-ahead forecasts. This result can also be seen in Fradkin (2007),

which attributes this to the presence of a constant used to fit the model to the early part of the

sample which causes model performance to suffer during periods of lower variance. Once we

enter the second half of the testing period, this bias decreases as we enter a period of higher

overall variance. For some stocks, there is even a period where the models under-predict equity

realized variance. Similar to how the constant was too high in the earlier part of our sample, the

constant term which caused a bias towards over-predicting equity variance in periods of lower

market stress is now too low to accurately predict the level of variance. This is especially true for

the bank stocks because during the second half of the sample, the financial sector was hit the

hardest by the turmoil in the mortgage backed securities market. With the recent turmoil in the

financial markets, it seems that authors of further work on HAR-RV type models using data from

2008 have to be especially cognizant of the constant term’s impact on the accuracy of predictions

of realized variance.

The performance of the models introduced in this paper in periods of high volatility also

requires greater investigation. Conveniently we are in the middle of one such period: since

September 2008, six of the ten largest daily percentage gains and five of the ten largest daily

percentage losses in the S&P 500 have occurred. Unfortunately, datasets used for this paper only

included price data up to 2007 but it is still possible to discuss what results we would expect had

more recent data been included. Since the summer of 2007, volatility in the stock prices of banks

24

has been driven by uncertainty over how to value the assets held on their balance sheets. This

uncertainty has increased the volatility of stocks in other sectors for a variety of reasons that

range from concerns about how declining credit availability will impact corporate earnings to

institutions selling stock in order to meet demands for cash. We would expect that for banking

stocks, sector variance effects would become more prominent because there is sector wide

uncertainty about the health of American financial institutions. In other words, investors are less

concerned about asynchronous company risk and more concerned about sector wide risk.

Conversely, we would expect that for pharmaceutical stocks, the market variable has a larger role

in determining equity variance because broader market events are playing a greater role in

determining returns on pharmaceutical stocks. For example, it could be that market participants

are simply unwilling to stomach the large swings in the stock market and instead choose to hold

other assets, or that funds are forced to sell their holdings of pharmaceutical stocks in order to

meet investor redemption requests. Overall we would also expect the results to be less accurate

because the model was calibrated using data from a period of much lower volatility.

5.5 Jump Test Results

In addition to investigating sector specific effects on stock variance through the HAR-RV

model, possible sector specific effects on stock price jumps were also investigated. This section

of the paper attempts to find sector effects for equity variance by examining periods of shared

jump days between companies in the same industry and attempting to link these shared jumps to

a possible news event. Jump days were examined for the pharmaceuticals industry in 2007 and

the Wall Street Journal was used to determine if there were any shared jump days that could be

attributed to sector-related news events. Summary statistics for the pharmaceuticals sector are

provided in Table 14.

25

Because pharmaceutical companies are often in direct competition with each other, we

might expect that a significant jump in one stock is associated with significant jumps in other

stocks in the sector. Three clusters of shared jump days were found: one from January 29th

-31st,

another in February 14th

, and one last cluster October 16th

-17th

. In each case, it was difficult to

discern any news that would likely affect the entire industry. Two events could have motivated

increases in sector wide jump measures: for the first cluster of jumps, the Thai government ruled

that they would sell special generic versions of drugs made by Abbot Labs and Bristol-Myers.

During that period, Abbot Labs, Merck, and Pfizer all had significant jumps. For the second

cluster, a European pharmaceuticals company released earnings and there were rumors that

Bristol-Myers would be acquired. On that day, both Bristol-Myers and Pfizer underwent

significant jumps.

The results show that it is difficult to investigate sector wide effects using the flagged

jump days. First, statistically significant jumps should reflect new and important information

entering the market, or else jumps would already be priced in. It is difficult to determine to what

extent these news items were surprises for the market. It is also difficult to isolate exactly what

motivates these jumps by simply looking for Wall Street Journal articles from that day. Finally,

there could be sector wide effects; however, they might not be strong enough to cause

statistically significant jumps. This qualitative process is neither elegant nor mathematically

satisfying, as we cannot quantify these sector effects on equity variance.

6. Conclusion

This paper has examined the inclusion of sector and market realized variance measures in

the HAR-RV model in order to better understand the influence of sector and market variance on

26

a specific stock’s variance. The intuition behind this was provided by popular models such as the

CAPM, which states that a stock’s returns are affected by non-company specific factors. This

paper has determined that the variance of a stock’s returns is also affected by non-company

specific factors. These factors were identified by first creating a model that included all relevant

measures of sector and market variance, then paring down that model using relevant statistical

methods. The results demonstrate that sector and market variances do play a role in determining

a stock’s variance, especially within the one week time horizon. Yesterday’s sector variance and

the daily deviation in the market variance from the previous month’s average both consistently

impact a stock’s variance.

In-sample and out-of-sample results also show that models which incorporate measures

of sector and market realized variance do a better job of forecasting variance in normal market

conditions. The inclusion of sector and market variance measures lead to consistent increases in

R-squared and decreases in mean squared errors. Overall, these results hold across both the

banking sector and the pharmaceuticals sector. However, in periods where there are high levels

of stock volatility, such as the past few months, these results might not hold. While the HAR-RV

model is a tremendously useful tool to forecast equity variance, this paper has demonstrated that

the basic model can generally be improved by the inclusion of measures of sector and market

variance. This is an important development because currently most models that forecast equity

variance only focus on measures of equity variance. These results suggest that there needs to be

additional research done on incorporating measures of sector and market variance into models of

equity variance. Further, the performance of predictive models of equity variance in periods of

high market stress also requires additional investigation.

27

7. Tables

Tables 1 – 3 correspond to the Full model and equation 11

Table 1: Coefficient Estimates and Significance Levels Day-Ahead

ABT BMY JNJ MRK PFE

βCD 0.1664*** 0.0422 0.1046* 0.1556*** 0.1328**

βCW 0.2039** 0.0453 -0.0826 0.0315 0.0055

βCM 0.5038*** 0.5735*** 0.4859** 0.1563 0.5300***

βSD 0.2420* 0.5398*** 0.3010*** 0.1822* 0.2283***

βSW -0.059 -0.0076 0.1991* 0.1216 0.0945

βSM -0.0535 -0.1063 -0.1956 0.1979 -0.0844

βSPD 0.2561 0.1185 0.1228 0.3043* 0.3788***

βSPW 0.0392 0.3303 0.5201 -0.0572 0.2057

βSPM -0.4811** -0.5143 -0.5261* -0.3814* -0.6540*

β0 .0000198 0.000005 -

0.0000009 0.000041** 0.00004***

* p<0.05, **p<0.01, ***p<0.001

Table 2: Coefficient Estimates and Significance Levels Week-Ahead

ABT BMY JNJ MRK PFE

βCD 0.0902** -0.0141 0.0493 0.0789** -0.00046

βCW 0.1854 0.0724 -0.0733 -0.04395 0.1308

βCM 0.6272*** 0.6081** 0.5064* 0.0820 0.4728**

βSD 0.1036 0.2397** 0.1907*** 0.0454 0.1990**

βSW 0.0473 0.2747 0.3081* 0.3411 0.0496

βSM 0.0214 -0.0382 -0.1060 0.3472 0.1411

βSPD 0.1492* 0.2371* 0.2697* 0.1441* 0.2328*

βSPW 0.0766 0.0110 0.0545 0.0294 0.1481

βSPM -0.6822** -0.6222 -0.4817* -0.5288* -0.7767**

β0 0.00004* 0.0000196 -0.000009 0.00006*** 0.00006***

* p<0.05, **p<0.01, ***p<0.001

28

Table 3: Coefficient Estimates and Significance Levels Month-Ahead

ABT BMY JNJ MRK PFE

βCD 0.0473** -0.0099 0.0115 0.0208 0.0078

βCW 0.1083* 0.1307 -0.0596 -0.0220 0.1779*

βCM 0.8062*** 0.4909* 0.4526 -0.1349 0.2707*

βSD 0.0627 0.1771*** 0.1190*** 0.0766* 0.1113***

βSW 0.0634 0.2053 0.2549 0.2130 -0.0524

βSM 0.1242 0.2219 0.1615 0.7081** 0.6033**

βSPD 0.0678 0.1242* 0.1110* 0.0785* 0.1046*

βSPW 0.0074 -0.0965 -0.0545 -0.0577 0.0003

βSPM -1.1189*** -0.9008*** -0.6644** -0.7346* -1.064***

β0 0.000053** 0.0000562* 0.000033* 0.00009*** 0.0000841

* p<0.05, **p<0.01, ***p<0.001

Table 4: In-sample R-Squared (1 day ahead)

R-Squared ABT BAC BK BMY C JNJ JPM MRK PFE USB WFC

HAR-RV 0.5133 0.5178 0.3331 0.4264 0.4788 0.4538 0.4739 0.2578 0.3511 0.5149 0.4139

S&P Only 0.5285 0.5219 0.3439 0.4525 0.4909 0.5222 0.5413 0.2766 0.3911 0.588 0.4733

Sector Only 0.528 0.5363 0.3496 0.4696 0.49 0.5407 0.5525 0.2792 0.3799 0.5941 0.5167

Full 0.5359 0.5403 0.3513 0.4719 0.4954 0.5563 0.5636 0.2884 0.3985 0.6142 0.525

Parsimony 0.5352 0.5342 0.3522 0.4706 0.4919 0.5417 0.5474 0.2871 0.3975 0.6112 0.5102

% Increase Base to Full 2.26% 2.25% 1.82% 4.55% 1.66% 10.25% 8.97% 3.06% 4.74% 9.93% 11.11%

S&P Only 1.52% 0.41% 1.08% 2.61% 1.21% 6.84% 6.74% 1.88% 4.00% 7.31% 5.94%

Sector Only 1.47% 1.85% 1.65% 4.32% 1.12% 8.69% 7.86% 2.14% 2.88% 7.92% 10.28%

% Loss Full to Pars -0.07% -0.61% 0.09% -0.13% -0.35% -1.46% -1.62% -0.13% -0.10% -0.30% -1.48%

29

Table 5: In-sample R-Squared (Week ahead)

R-Squared ABT BAC BK BMY C JNJ JPM MRK PFE USB WFC

HAR-RV 0.6209 0.6366 0.5187 0.604 0.532 0.5225 0.4353 0.3483 0.4684 0.622 0.5958

S&P Only 0.6332 0.6413 0.5274 0.6336 0.5497 0.6 0.5602 0.3672 0.5015 0.6493 0.6206

Sector Only 0.628 0.6641 0.5355 0.6399 0.5558 0.6195 0.5872 0.4008 0.4987 0.6637 0.642

Full 0.6369 0.6681 0.5355 0.6481 0.5623 0.6427 0.6141 0.4095 0.5161 0.668 0.6463

Parsimony 0.632 0.6576 0.5358 0.6435 0.5532 0.6333 0.5792 0.3843 0.5112 0.6654 0.6376

% Increase Base to Full 1.60% 3.15% 1.68% 4.41% 3.03% 12.02% 17.88% 6.12% 4.77% 4.60% 5.05%

S&P Only 1.23% 0.47% 0.87% 2.96% 1.77% 7.75% 12.49% 1.89% 3.31% 2.73% 2.48%

Sector Only 0.71% 2.75% 1.68% 3.59% 2.38% 9.70% 15.19% 5.25% 3.03% 4.17% 4.62%

% Loss Full to Pars -0.49% -1.05% 0.03% -0.46% -0.9% -0.94% -3.49% -2.5% -0.49% -0.26% -0.87%

Table 6: In-sample R-Squared (Month ahead)

R-Squared ABT BAC BK BMY C JNJ JPM MRK PFE USB WFC

HAR-RV 0.5992 0.5853 0.482 0.5927 0.4984 0.4624 0.4269 0.3437 0.4609 0.6295 0.6514

S&P Only 0.6305 0.5886 0.4914 0.6117 0.5129 0.4919 0.5042 0.354 0.4771 0.646 0.6656

Sector Only 0.6036 0.5916 0.5006 0.6115 0.5244 0.5234 0.4971 0.4368 0.472 0.6428 0.666

Full 0.6364 0.6023 0.5016 0.6277 0.5338 0.5482 0.5184 0.4663 0.511 0.6515 0.6764

Parsimony 0.603 0.5898 0.4989 0.6106 0.516 0.5186 0.5007 0.3996 0.4754 0.6442 0.6642

New 22 0.6355 0.6004 0.5007 0.6252 0.5221 0.5383 0.4992 0.4605 0.5084 0.6472 0.6746

% Increase

Base to Full 3.72% 1.70% 1.96% 3.50% 3.54% 8.58% 9.15% 12.26% 5.01% 2.20% 2.50%

S&P Only 3.13% 0.33% 0.94% 1.90% 1.45% 2.95% 7.73% 1.03% 1.62% 1.65% 1.42%

Sector Only 0.44% 0.63% 1.86% 1.88% 2.60% 6.10% 7.02% 9.31% 1.11% 1.33% 1.46%

% Loss Full to Pars -3.34% -1.25% -0.27% -1.71% -1.78% -2.96% -1.77% -6.67% -3.56% -0.73% -1.22%

Full to Red 22 -0.09% -0.19% -0.09% -0.25% -1.17% -0.99% -1.92% -0.58% -0.26% -0.43% -0.18%

30

Table 7: Out-of-sample Mean Squared Errors 1 Day Ahead

ABT BAC BK BMY C JNJ JPM MRK PFE USB WFC

Full Base 0.808 1.33 8.8 1.75 2.89 0.146 4.44 0.918 0.924 2.12 2.29

Full 0.733 1.17 8.87 1.54 2.9 0.285 4.72 0.843 0.925 2.46 2.3

Pars 0.734 1.18 8.87 1.56 2.96 0.217 4.95 0.85 0.924 2.34 2.24

Early Base 0.455 0.534 0.528 1.08 0.621 0.114 0.526 0.528 1.47 0.263 0.292

Full 0.427 0.436 0.411 0.982 0.544 0.124 0.422 0.457 1.34 0.337 0.255

Pars 0.429 0.431 0.415 0.985 0.56 0.115 0.415 0.477 1.34 0.338 0.242

Late Base 1.16 2.15 17.2 2.42 5.19 0.177 8.41 1.3 0.384 4.02 4.31

Full 1.03 1.92 17.5 2.09 5.3 0.444 9.08 1.22 0.517 4.61 4.37

Pars 1.04 1.93 17.5 2.13 5.39 0.318 9.55 1.22 0.517 4.38 4.28

All values are to the 10^-8 power

Table 8: Out-of-sample Mean Squared Errors 5 Days Ahead

ABT BAC BK BMY C JNJ JPM MRK PFE USB WFC

Full Base 0.535 0.936 5.32 0.882 1.66 0.129 2.33 0.56 0.487 1.53 1.42

Full 0.473 0.909 5.19 0.746 1.67 0.183 2.26 0.435 0.452 1.52 1.4

Pars 0.501 0.905 5.21 0.748 1.75 0.182 2.28 0.463 0.499 1.48 1.37

Early Base 0.302 0.361 0.518 0.428 0.596 0.124 0.723 0.365 0.638 0.29 0.199

Full 0.281 0.305 0.457 0.322 0.382 0.101 0.417 0.233 0.553 0.312 0.174

Pars 0.293 0.273 0.465 0.334 0.455 0.102 0.342 0.288 0.564 0.308 0.156

Late Base 0.766 1.52 10.2 1.33 2.73 0.135 3.96 0.753 0.339 2.8 2.65

Full 0.661 1.52 9.99 1.16 2.98 0.264 4.14 0.634 0.352 2.75 2.64

Pars 0.705 1.55 10 1.16 3.06 0.261 4.24 0.635 0.436 2.67 2.6

All values are to the 10^-8 power

31

Table 9: Out-of-sample Mean Squared Errors 22 Days Ahead

ABT BAC BK BMY C JNJ JPM MRK PFE USB WFC

Full Base 0.588 1.37 6.19 0.976 2.84 0.252 3.21 0.616 0.579 2.04 2.1

Full 0.45 1.37 5.86 0.769 2.98 0.168 3 0.457 0.445 1.94 2.06

Pars 0.572 1.35 5.95 0.873 2.89 0.264 3.04 0.512 0.601 1.96 2.06

Red. 22 0.454 1.35 5.88 0.772 3.05 0.156 3.07 0.465 0.445 1.96 2.05

Early Base 0.338 0.452 1.01 0.421 0.878 0.252 1.27 0.457 0.669 0.561 0.188

Full 0.279 0.416 0.93 0.304 0.399 0.155 0.804 0.191 0.577 0.603 0.186

Pars 0.336 0.371 0.947 0.323 0.651 0.198 0.761 0.335 0.64 0.548 0.159

Red. 22 0.277 0.408 0.927 0.302 0.407 0.155 0.816 0.191 0.581 0.612 0.189

Late Base 0.834 2.3 11.5 1.52 4.83 0.252 5.18 0.773 0.491 3.55 4.04

Full 0.617 2.34 10.9 1.23 5.6 0.182 5.23 0.718 0.316 3.29 3.95

Pars 0.805 2.35 11 1.41 5.17 0.328 5.36 0.687 0.563 3.39 4

Red. 22 0.627 2.3 10.9 1.24 5.74 0.157 5.37 0.735 0.312 3.34 3.95

All values are to the 10^-8 power

Table 10: F-test P-value for Sector Weekly, Monthly and Market Weekly

Days Days

ABT 1 0.8995 JPM 1 0.0117

5 0.7719 5 0.3126

22 0.7116 22 0.5838

BAC 1 0.7636 MRK 1 0.072

5 0.796 5 0.0055

22 0.3077 22 0.0005

BK 1 0.8014 PFE 1 0.6807

5 0.693 5 0.3267

22 0.5433 22 0.0128

BMY 1 0.1968 USB 1 0.2501

5 0.4718 5 0.8061

22 0.2776 22 0.5671

C 1 0.074 WFC 1 0.1319

5 0.2568 5 0.7033

22 0.0408 22 0.8724

JNJ 1 0.0205

5 0.0295

22 0.0767

32

Tables 11 – 12 correspond to the Parsimonious model and equation 16

Table 11: Coefficient Estimates and Significance Levels Day-Ahead (Parsimonious Model)

ABT BMY JNJ MRK PFE

βCD 0.1829*** 0.0464 0.1152* 0.1227* 0.1183**

βCW 0.2020*** 0.1003 0.1958* 0.0648 0.1173*

βCM 0.3689*** 0.4219*** 0.1914** 0.3268*** 0.3933***

βSD 0.1968* 0.5258*** 0.3681*** 0.2794*** 0.2573***

βSPDiff 0.2793 0.2019* 0.2165* 0.2742* 0.4282***

β0 0.0000194* -0.000002 -0.000005 0.00005*** 0.00004***

* p<0.05, **p<0.01, ***p<0.001

Table 12: Coefficient Estimates and Significance Levels 5 Days Ahead (Parsimonious Model)

ABT BMY JNJ MRK PFE

βCD 0.0845** -0.0367 0.0312 -0.0081 -0.0159

βCW 0.2629*** 0.2266*** 0.1620 0.1187 0.2066**

βCM 0.4340*** 0.4798*** 0.3238** 0.3225*** 0.4038***

βSD 0.1040* 0.3055*** 0.2852*** 0.2750*** 0.2209**

βSPDiff 0.1773* 0.2615** 0.2697* 0.1178* 0.2710*

β0 0.000036** 0.000025** 0.000015 0.00007*** 0.00006***

* p<0.05, **p<0.01, ***p<0.001

Table 13 corresponds to the Reduced Monthly model and equation 17

Table 13: Coefficient Estimates and Significance Levels 22 Days Ahead (Reduced Monthly Model)

ABT BMY JNJ MRK PFE

βCD 0.0473** -0.0367 -0.0107 0.0019102 0.0137

βCW 0.1520*** 0.2189*** 0.1132* 0.0854 0.1490**

βCM 0.7654*** 0.4259* 0.2889 -0.2257 0.2902*

βSD 0.0997** 0.2836*** 0.2204*** 0.1529** 0.1366**

βSM 0.1556 0.3354 0.3276 0.8596** 0.5318**

βSPM -1.053*** -0.8973** -0.6170** -0.7440* -0.9651***

β0 0.000053* 0.000056* 0.000033* 0.00009*** 0.00008***

* p<0.05, **p<0.01, ***p<0.001

33

Table 14: Pharmaceutical Jump Days (1997 - 2007)

ABT BMY JNJ MRK PFE

Mean RV (x10-4

) 2.8404 3.3421 1.8252 2.429 2.8078

Mean BV (x10-4

) 2.603 2.9993 1.6801 2.232 2.6035

Number of Jump Days 110 137 114 86 95

34

8. Figures

Figure 1: JP Morgan (JPM) Daily Realized Variance July 98 – Jan 08

35

Figure 2: JPM Day Ahead Out-of-sample Forecasts and Residuals

36

Figure 3: JPM 5 Days Ahead Out-of-sample Forecasts and Residuals

37

Figure 4: JPM 22 Days Ahead Out-of-sample Forecasts and Residuals

38

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